This paper proposes a local linear estimator for diurnal patterns of transaction durations under a special nonparametric regression model, whose asymptotics are different to any known results. An iterative plug-in algorithm is developed for selecting the bandwidth. The ACD model is then applied to analyze the standardized durations. Data examples show that the proposals work well in practice.Autoregressive conditional duration, diurnal duration patterns, local linear estimator, bandwidth selection, iterative plug-in.

Yuanhua Feng

English

Research Papers in Economics

CENTER FOR INTERNATIONAL ECONOMICS
Working Paper Series
Working Paper No. 2011-11
Data-driven estimation of diurnal duration patterns
Yuanhua Feng
December 2011Data-driven estimation of diurnal duration patterns
Yuanhua Feng
University of Paderborn
Abstract
This paper proposes a local linear estimator for diurnal patterns of transaction
durations under a special nonparametric regression model, whose asymptotics are
dierent to any known results. An iterative plug-in algorithm is developed for se-
lecting the bandwidth. The ACD model is then applied to analyze the standardized
durations. Data examples show that the proposals work well in practice.
Keywords: Autoregressive conditional duration, diurnal duration patterns, local
linear estimator, bandwidth selection, iterative plug-in.
JEL Codes: C14, C41
1 Introduction
Let t0 < t1 < ::: < tN < tN+1 = T be the time points at which trades occur and
xi = ti ti 1 the duration between trades, where t0 = 0 and N is the (random) number of
trades. Consider the model xi = i i"i, where i is a (deterministic) diurnal pattern and
 i is the conditional mean after removing i. Estimation of (t) is a necessary step for
further econometric analysis of transaction durations using e.g. the ACD (autoregressive
conditional duration) of Engle and Russell (1998). Dierent approaches for estimating
(t) are introduced in the literature. For instance, Engle and Russell (1998) propose to
use a cubic spline and a nonparametric estimator was introduced by Veredas et al. (2002).
For further ideas on the estimation of  see e.g. Bauwens and Giot (2000), Hautsch (2004)
and references therein.
This paper focuses on developing a data-driven local linear estimator of . For this
purpose a nonparametric regression model with some special features is dened, where 
is the mean function and the scale function at the same time, and is also proportional to
the reciprocal of the design density. The asymptotic variance and asymptotically optimal
1bandwidth of ^  are hence dierent to any known results. An iterative plug-in (IPI, Gasser
et al., 1991) algorithm with some improvements is developed for selecting the bandwidth
under the current model. Data examples show that the proposals work well in practice.
2 Local linear estimation of the diurnal pattern
The model xi = i i"i (Engle and Russell, 1998, 2010) can be rewritten as follows:
xi = (ti) + (ti)(yi   1);i = 1;:::;N; (1)
where (ti) is the local mean of xi, yi =  i"i with E(yi) = 1,  i is the conditional mean
of yi, which can be modeled by a unit ACD with  i = ! +
Pp
j=1 jyi j +
Pq
k=1 k i k,
! > 0, j;k  0 and ! = 1  
P
j  
P
k, and "i are i.i.d. non-negative random
variables with E("i) = 1. Let i = yi  1. Model (1) is a special nonparametric regression
with stationary errors i. 1: (t) is its mean and scale function at the same time; 2.
xi = ti   ti 1 is determined by the regressor and 3. (ti) ! 0, as N ! 1. Moreover, let
f(t) > 0 be the design density of t and m(t) = [f(t)] 1, we have N(ti) = m(ti)+o(N 1).
These features will aect the asymptotic properties of ^ (t) strongly. We see m(t) is a
given function, but (t) depends on N. Hence we will rst estimate m(t) from zi = Nxi
and then put ^ (t) = ^ m(t)=N, since the discussion with ^ m(t) is more convenient.
The derivatives m()(t) can be estimated by minimizing the weighted least squares
Q =
N X
i=1
fzi   a0(t)   a1(t)(ti   t)   :::   ap(t)(ti   t)
pg
2 K

ti   t
b

; (2)
where K is a kernel and b is the bandwidth. We have ^ m()(t) = !^ a and ^ ()(t) =
!^ a=N. In the sequel, local linear estimators ^ m(t) and ^ (t) = ^ m(t)=N will be used. Let
^ yi = xi=^ (ti). A suitable ACD model can be tted to ^ yi using the standard QML method.
Let var[^ m(t)], B[^ m(t)], var[^ (t)] and B[^ (t)] denote the variance and bias of ^ m(t)
and ^ (t), respectively. We have var[^ (t)] = var[^ m(t)]=N2 and B[^ (t)]  B[^ m(t)]=N.
Asymptotic properties of ^ m(t) can be derived under regularity conditions. The asymptotic
bias of ^ m(t) is the same as for a common local linear estimator. Let (k) = cov(yi;yi+k),
S =
P
(k) and R(K) =
R
K2(u)du. By adapting known results in the literature, we can
2obtain the asymptotic variance of ^ m(t) at 0 < t < T:
var[^ m(t)] 
R(K)S
Nbf(t)
m
2(t) =
R(K)S
Nb
m
3(t); (3)
which is proportional to m3(t). Furthermore, let I(K) =
R
u2K(u)du. The optimal
bandwidth for estimating m(t) and (t) on [0;T], bopt, is the same and is asymptotically
bopt  bA =

R(K)S
I2(K)
R
m3(t)dt R
[m00(t)]2dt
1=5
N
 1=5: (4)
Let V = R(K)S
R
m3(t)dt and b = o(bA). At an interior point b < t < T   b, we have
p
Nb[^ m(t)   m(t)]
D  ! N(0;V ): (5)
3 Bandwidth selection
An IPI algorithm based on (4) with a starting bandwidth b0 = T=10 is developed, which
extends the idea of Gasser et al. (1991) to random design nonparametric regression with
dependent errors, huge sample size and very large error variance, where S is estimated
by a lag-window estimator with the maximal lag K  4N1=3 (see B uhlmann, 1996 and
Feng and Wang, 2011). For any reasonable b0, the IPI bandwidth converges to the same
x-point. The above choice of b0 ensures that S can already be well estimated at the
beginning. By an IPI algorithm, m00(t) in the jth iteration is estimated using a bandwidth
b2j calculated from bj 1. Here, the formula b2j = T(bj=T)1=2 (Beran and Feng, 2002) is
used, which works better than ~ b2j = bj 1  N1=10 (Gasser et al., 1991) in the current
case. The choice of b0 and b2j also helps to reduce a few iterations. Starting with b0, the
algorithm reads as follows:
i) In the jth iteration, obtain ^ j(t) with bj 1. Let ^ yji = xi=^ j(ti). Calculate ^ j(k) from
^ yji and let ^ Sj =
P
jkj<K
wk^ j(k), where wk are the Bartlett-weights and K  4N1=3.
ii) Obtain ^ mj(ti) with bj 1 and ^ m00
j(ti) using local cubic with b2j = T(bj 1=T), where
 = 1=2 or  = 5=7. Let ^ Ij(m3) = 1
N
N P
i=1
[^ mj(ti)]3 and ^ Ij([m00]2) = 1
N
N P
i=1
[^ m00
j(ti)]2.
iii) Improve bj 1 by bj =

R(K)^ Sj
I2(K)
^ Ij(m3)
^ Ij([m00]2)
1=5
N 1=5 and increase j by one.
3iv) Repeatedly carry out i) to iii) until convergence is reached and put ^ bA = bj.
Under regularity assumptions, in particular that E(y4
i) < 1, the rate of convergence of ^ bA
is the same as for nonparametric regression with independent errors and is determined by
. If  = 1=2, ^ bA is most stable so that ^ bA  bopt[1+O(N 1=5)+Op(N 1=2)]. For  = 5=7,
^ bA achieves the highest rate of convergence with ^ bA  bopt[1+O(N 2=7)+Op(N 2=7)] (see
Lemma 1 of Beran and Feng, 2002 with  = 0). Theoretically, the use of  = 2=7 is more
preferable, but the error variance in the current case is very large and the stability of ^ bA
is more important. Hence  = 1=2 will be used in this paper. Furthermore, we propose to
calculate the nal estimates using the local bandwidths ^ b(t) = ^ bA  ^ m(t). This is a quasi
k-NN method and leads to further improvement of the estimation quality.
4 Practical implementation and applications
An R code is written by means of the \lm" function with proper weights, which is applied
to stock durations of a few German rms (T = 8:5 and b0 = 0:85 hours). It shows that
the algorithm works well in practice. Bandwidths (in hours) selected with the bisquare
kernel for BMW stock durations in the rst two weeks in Aug 2011 are given in Table
1 as examples. Note that ^ bA is jointly determined by N, S and the complexity of (t)
Table 1. Selected bandwidths (h) for BMW diurnal duration patterns in two weeks
1.8.11 2.8.11 3.8.11 4.8.11 5.8.11 8.8.11 9.8.11 10.8.11 11.8.11 12.8.11
N 9249 14527 12789 19477 22317 24377 34286 25995 19437 15404
^ bA 1.0657 1.4086 0.9794 0.8611 1.0867 0.5565 1.3248 0.8810 0.8397 0.9602
and changes strongly from one day to another. The largest bandwidth is ^ bA = 1:4086
on Aug 2, and the smallest is ^ bA = 0:5565 on Aug 8, 2011. The former is much bigger
and the latter is much smaller than b0. The number of iterations required also varies
from case to case. It is e.g. just 3 on Aug 1, but 15 on Aug 9, 2011. The observed
durations, the estimated diurnal patterns, ^ (t), and the standardized durations, ^ yi, on
Aug 2 and 8, 2011, respectively, are shown in Figure 1. The diurnal pattern on Aug 2,
2011 has a typical inverse U-form except for a sub-peak at about one hour prior to the
close. In this case the smoothing quality of ^ (t) is very good. The diurnal pattern on
Aug 8, 2011 looks very special with two peaks in the morning and in the afternoon and
4some minor structures. Hence ^ bA is now very small. The bandwidths used around lunch
time and prior to the close seem to be not large enough. This problem might be solved
using a varying bandwidth taking ^ m00(t) into account. Discussion on this is beyond the
aim of this paper. The standardized durations seem to be stationary with clear clusters,
except for a few outliers. Two EACD(1, 1) (exponential ACD) models are tted to ^ yi in
Figures 1(c) and 1(f), respectively, using the \fACD" package in R (Perlin, 2009). The
results are listed in Table 2. These models t the data very well. The estimation quality
Table 2. EACD parameters for the 1. (left) and 2. (right) examples with p-values
!   !  
.0362(.0030) .1197(.0053) .8552(.0067) .0487(.0028) .1120(.0039) .8473(.0054)
after removing (t) is improved. For instance, the EACD model obtained from xi on Aug
2, 2011, is non-stationary with ^  + ^   1. However, the rst model given in Table 2
estimated from ^ yi does not share this problem.
5 Final remarks
A local linear estimator of diurnal duration patterns on nancial markets with special
asymptotics is proposed. An IPI bandwidth selector is developed. Applications show
that the proposal works well in practice. To our knowledge this paper is the rst eort to
develop a data-driven estimator of diurnal duration patterns. There are still many open
questions in this context, such as further improvements of the data-driven algorithm, the
estimation of m(t) via ^ f(t) and the signicance test of ^ (t). Furthermore, discussion on
the application of the estimation results and detailed economic sources for a non-typical
diurnal duration dynamics like those happened on Aug 8, 2011 is also of great interest.
Acknowledgment
We are grateful to Mr Christian Peitz and Mr Kan Wang for preparing the data, useful
discussion and further application of the proposed algorithm.
5References
Bauwens, L. and Giot, P. (2000). The logarithmic ACD model: An application to the
bid/ask quote process of two NYSE stocks. Ann. d'Econ. et de Stat., 60, 117-149.
Beran, J. and Y. Feng (2002). Iterative plug-in algorithms for SEMIFAR models - deni-
tion, convergence and asymptotic properties. J. Comput. Graph. Stat. 11, 690-713.
B uhlmann, P. (1996). Locally adaptive lag-window spectral estimation. J. Time Ser.
Anal., 17, 247-270.
Engle, R.F. and J.R. Russell (1998). Autoregressive conditional duration: a new model
for irregularly spaced transaction data. Econometrica, 66, 1127-1162.
Engle, R.F. and J.R. Russell (2010). Analysis of high frequency data. In Handbook of
Financial Econometrics, Vol. 1, 383-426, ed. by Y. Ait-Sahalia and L.P. Hansen, Elsevier.
Feng, Y. and Wang, K. (2011). Modelling long term and conditional dynamics in average
durations by a semiparametric ACD. Forthcoming preprint, University of Paderborn.
Gasser, T., A. Kneip & W. K ohler (1991) A exible and fast method for automatic
smoothing. J. Amer. Statist. Assoc. 86, 643{652.
Hautsch, N. (2004). Modelling irregularly spaced nancial data. Springer, Berlin.
Perlin, M. (2009). Autoregressive conditional duration models in R. On R-Forge.
Veredas, D., Rodriguez-Poo, J., and Espasa, A. (2002). On the (intradaily) seasonality,
dynamics and durations zero of a nancial point process. CORE Disc. Paper, 2002/23.
610 12 14 16
0
2
0
4
0
6
0
8
0
1
0
0
1
4
0
Trading hours
(a) BMW durations on Aug 2, 2011 (Sec.)
10 12 14 16
0
2
0
4
0
6
0
8
0
1
2
0
Trading hours
(d) BMW durations on Aug 8, 2011 (Sec.)
10 12 14 16
0
.
5
1
.
0
1
.
5
2
.
0
2
.
5
3
.
0
3
.
5
Trading hours
(b) Diurnal pattern on Aug 2, 2011 (Sec.)
10 12 14 16
0
.
6
0
.
8
1
.
0
1
.
2
1
.
4
1
.
6
1
.
8
Trading hours
(e) Diurnal pattern on Aug 8, 2011 (Sec.)
10 12 14 16
0
2
0
4
0
6
0
8
0
1
0
0
1
2
0
Trading hours
(c) Standardized durations on Aug 2, 2011
10 12 14 16
0
2
0
4
0
6
0
8
0
1
0
0
Trading hours
(f) Standardized durations on Aug 8, 2011
Figure 1: Observed durations (above), estimated diurnal patterns (middle) and standard-
ized durations (bottom) of BMW stock on two days.
7Recent discussion papers 
 
2011-11  Yuanhua Feng  Data-driven estimation of diurnal duration patterns 
     
2011-10  B. Michael Gilroy 
Birke Thuy Duong Nguyen 
Ist Fairer Handel Wirklich Fair? 
     
2011-09  Yuanhua Feng 
Zhichao Guo 
Christian Peitz 
Xiangyong Tan 
A tree-form constant market share model for growth causes   
in international trade based on multi-level classification 
     
2011-08  Tim Krieger 
Daniel Meierriecks 
Ökonomische Aspekte des neuen globalen Terrorismus 
     
2011-07  Tim Krieger 
Daniel Meierrieks 
Terrorist Financing and Money Laundering 
     
2011-06  Thomas Gries 
Stefan Jungblut 
Wim Naudé 
New Firm Creation and Failure: A Matching Approach 
     
2011-05  Daniel Kaimann  "To infinity and beyond!" - A genre-specific film analysis  
of movie success mechanisms 
     
2011-04  Yuanhua Feng 
Zhichao Guo 
Christian Peitz 
Xiangyong Tan 
A tree-form constant market share analysis for modelling 
growth causes in international trade 
     
2011-03  Zhichao Guo 
Yuanhua Feng 
Xiangyong Tan 
Impact of China’s accession to WTO and the financial crisis   
on China’s exports to Germany 
     
2011-02  Alexander Haupt 
Tim Krieger 
Thomas Lange 
Competition for the International Pool of Talent: Education 
Policy and Student Mobility 
     
2011-01  B. Michael Gilroy 
Daniel Kruse 
Die Prinzipal-Agent-Theorie als Erklärungsinstrumentarium 
von Korruption: Angewendet auf den Praxisfall „Siemens“ 
     
2010-09  Yuanhua Feng  An iterative plug-in algorithm for decomposing seasonal 
time series using the Berlin Method 
     
2010-08   Zhichao Guo 
 Yuanhua Feng 
 Xiangyong Tan 
Short- and long-term impact of remarkable economic events 
on the growth causes of China-Germany trade in agri-food 
products 
[published in: Economic Modelling 28 (2011), 2359-2368] 
     
2010-07  B. Michael Gilroy 
Elmar Lukas 
Christian Heimann 
Welchen Einfluss hat die Anwesenheit von 
ausländischen und multinationalen Unternehmungen 
auf die deutschen Exporte? 
     
2010-06  Stefan Gravemeyer      
Thomas Gries              
Income and disparity in Germany and China 
     
2010-05  Thomas Gries               
Margarete Redlin 
Short-run and Long-run Dynamics of Growth, 
Inequality and Poverty in the Developing World 
     
2010-04  Stefan Gravemeyer      
Thomas Gries             
Poverty in Shenzen Jinjun Xue 
     
2010-03  Alexander Haupt 
Tim Krieger 
Thomas Lange 
A Note on Brain Gain and Brain Drain: 
Permanent Migration and Education Policy 
     
2010-02  Sarah Brockhoff 
Tim Krieger 
Daniel Meierrieks 
Ties That Do Not Bind (Directly): 
The Education Terrorism Nexus Revisited 
 
2010-01 
 
 
 
Claus-Jochen Haake, 
Tim Krieger, 
Steffen Minter 
 
 
On the institutional design of burden sharing when financing 
external border enforcement in the EU 
 
2009-06  Tim Krieger, 
Stefan Traub 
Wie hat sich die intragenerationale Umverteilung 
in der staatlichen Säule des Rentensystems verändert? 
Ein internationaler Vergleich auf Basis von LIS-Daten 
[published in: Jahrbücher für Nationalökonomie und Statistik 
231 (2011), 2, 266-287] 
     
2009-05  Karin Mayr, 
Steffen Minter, 
Tim Krieger 
Policies on illegal immigration in a federation [forthcoming 
in: Regional Science and Urban Economics] 
 
     
2009-04  Tim Krieger, 
Daniel Meierrieks 
Terrorism in the Worlds of Welfare Capitalism 
[published in: Journal of Conflict Resolution 54, 6, 902-939.] 
     
2009-03  Alexander Haupt, 
Tim Krieger 
The role of mobility in tax and subsidy competition 
     
2009-02  Thomas Gries, 
Tim Krieger, 
Daniel Meierrieks 
Causal Linkages Between Domestic Terrorism and  
Economic Growth 
[forthcoming in: Defense and Peace Economics] 
 
     
2009-01  Andreas Freytag, 
Jens J. Krüger, 
Daniel Meierrieks, 
Friedrich Schneider 
The Origin of Terrorism - Cross-Country Estimates on  
Socio-Economic Determinants of Terrorism 
     
2008-11  Thomas Gries, 
Magarete Redlin 
China’s provincial disparities and the determinants of pro-
vincial inequality 
[published in: Journal of Chinese economic and business 
studies 7 (2009), 2, 259-281.] 
     
2008-10  Thomas Gries, 
Manfred Kraft, 
Daniel Meierrieks 
Financial Deepening, Trade Openness and Economic 
Growth in Latin America and the Caribbean 
[forthcoming in: Applied Economics] 
     
2008-09 
 
Stefan Gravemeyer, 
Thomas Gries, 
Jinjun Xue 
Discrimination, Income Determination and Inequality – The 
case of Shenzen 
[forthcoming in: Urban Studies] 
     
2008-08 
 
Thomas Gries, 
Manfred Kraft, 
Daniel Meierrieks 
Linkages between Financial Deepening, Trade Openess 
and Economic Development: Causality Evidence from Sub-
Saharan Africa 
[published in: World Development 37 (2009), 1849-1860.] 
     
2008-07  Tim Krieger, 
Sven Stöwhase 
Diskretionäre rentenpolitische Maßnahmen und die Entwick-
lung des Rentenwerts in Deutschland von 2003-2008 [publi-
shed in: Zeitschrift für Wirtschaftspolitik 58 (2009), 1, 36-54.] 
     2008-06  Tim Krieger, 
Stefan Traub 
Back to Bismarck? Shifting Preferences for Intrageneration-
al Redistribution in OECD Pension Systems 
     
2008-05  Tim Krieger, 
Daniel Meierrieks 
What causes terrorism? 
[published in: Public Choice 147 (2010), 1-2, 3-27.] 
     
2008-04  Thomas Lange  Local public funding of higher education when 
students and skilled workers are mobile 
[published in: Finanzarchiv 65 (2009), 2, 178-199.] 
     
2008-03  Natasha Bilkic,  
Thomas Gries, 
Margarethe Pilichowski 
Stay at school or start working? - Optimal timing of leaving 
school under uncertainty and irreversibility 
 
 
2008-02  Thomas Gries, 
Stefan Jungblut, 
Tim Krieger, 
Henning Meier 
 
Statutory retirement age and lifelong learning 
2008-01  Tim Krieger, 
Thomas Lange 
 
Education policy and tax competition with imperfect student 
and labor mobility 
[published in: International Tax and Public Finance 17 
(2008), 6, 587-606.] 
 
2007-05  Wolfgang Eggert, 
Tim Krieger, 
Volker Meier 
Education, unemployment and migration 
[published in: Journal of Public Economics 94 (2010), 5-6, 
354-362.] 
     
2007-04  Tim Krieger, 
Steffen Minter 
Immigration amnesties in the southern EU member states - 
a challenge for the entire EU? 
[published in: Romanian Journal of European Studies  5-
6/2007, 15-32.] 
     
2007-03  Axel Dreher, 
Tim Krieger 
Diesel price convergence and mineral oil taxation in Europe 
[published in: Applied Economics 42 (2010), 15, 1955-1961] 
     
2007-02  Michael Gorski, 
Tim Krieger, 
Thomas Lange 
Pensions, education and life expectancy 
   
 
 
2007-01  Wolfgang Eggert, 
Max von Ehrlich, 
Robert Fenge, 
Günther König 
Konvergenz- und Wachstumseffekte der europäischen Re-
gionalpolitik in Deutschland 
[published in: Perspektiven der Wirtschaftspolitik 8 (2007), 
130-146.] 
     
2006-02  Tim Krieger  Public pensions and return migration 
[published in: Public Choice 134 (2008), 3-4, 163-178.] 
     
2006-01  Jeremy S.S. Edwards, 
Wolfgang Eggert,  
Alfons J. Weichenrieder 
The measurement of firm ownership and its effect on mana-
gerial pay 
[published under the title “Corporate Governance and Pay 
for Performance: Evidence from Germany” in: Economics of 
Governance 10 (2009), 1, 1-26.] 
 
 

Data-driven estimation of diurnal duration patterns

http://groups.uni-paderborn.de/fiwi/RePEc/pdf/wpaper/WP44.pdf

Data-driven estimation of diurnal duration patterns

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics